A name of a cocktail drink that starts with Q?
There doesn't seem to be many. On Wikipedia's listing of cocktails, the only Q is the Quentão, which means "very hot. It's a warm Brazilian cocktail mixing cachaca and spices. It's definitely a regional drink, as it's hard to even acquire cachaca outside of Brazil.
On the cocktail database Drinksmixer, this is where all of the Q cocktails are:
Lots of things starting with Queen or Quick, but nothing that'd I'd call well-known.